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\begin{document}

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\section*{Dedekind zeta function}

\hypertarget{context}{}\subsubsection*{{Context}}\label{context}

\hypertarget{arithmetic_geometry}{}\paragraph*{{Arithmetic geometry}}\label{arithmetic_geometry}

[[!include arithmetic geometry - contents]]

\hypertarget{contents}{}\section*{{Contents}}\label{contents}

\noindent\hyperlink{idea}{Idea}\dotfill \pageref*{idea} \linebreak
\noindent\hyperlink{properties}{Properties}\dotfill \pageref*{properties} \linebreak
\noindent\hyperlink{special_values}{Special values}\dotfill \pageref*{special_values} \linebreak
\noindent\hyperlink{adelic_integral_expression}{Adelic integral expression}\dotfill \pageref*{adelic_integral_expression} \linebreak
\noindent\hyperlink{functional_equation}{Functional equation}\dotfill \pageref*{functional_equation} \linebreak
\noindent\hyperlink{the_pole_and_the_class_number_formula}{The pole and the class number formula}\dotfill \pageref*{the_pole_and_the_class_number_formula} \linebreak
\noindent\hyperlink{RelationToThetaFunctions}{Relation to other zeta-, theta-, and L-functions}\dotfill \pageref*{RelationToThetaFunctions} \linebreak
\noindent\hyperlink{analogs_over_complex_curves}{Analogs over complex curves}\dotfill \pageref*{analogs_over_complex_curves} \linebreak
\noindent\hyperlink{function_field_analogy}{Function field analogy}\dotfill \pageref*{function_field_analogy} \linebreak
\noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak


\hypertarget{idea}{}\subsection*{{Idea}}\label{idea}

The \emph{Dedekind zeta function} is a generalization of the [[Riemann zeta function]] from the [[rational numbers]]/[[integers]] to [[number fields]]/their [[rings of integers]].

The analog for [[function fields]] is the [[Weil zeta function]], while the generalization to [[higher dimensional arithmetic geometry]] are the [[arithmetic zeta functions]].

\hypertarget{properties}{}\subsection*{{Properties}}\label{properties}

\hypertarget{special_values}{}\subsubsection*{{Special values}}\label{special_values}

For $K$ a [[number field]] then all [[special values of L-functions|special values]] of the Dedekind zeta function $\zeta_K(n)$ for integer $n$ happen to be [[periods]] (\href{http://mathoverflow.net/a/54244/381}{MO comment}).

Based on this (\hyperlink{Deligne79}{Deligne 79}) identified \emph{critical values} of $L$-functions (at certain integers) and conjectured that these are all these are algebraic multiples of [[determinants]] of [[matrices]] whose entries are [[periods]]. For more on this see at \emph{[[special values of L-functions]]}.

\hypertarget{adelic_integral_expression}{}\subsubsection*{{Adelic integral expression}}\label{adelic_integral_expression}

The Dedekind zeta function has an [[adelic integral]] expression in direct analogy to that of the [[Riemann zeta function]]. This is due to (\hyperlink{Tate50}{Tate 50}), highlighted by [[Ivan Fesenko]] in (\hyperlink{GoldfeldHundley11}{Goldfeld-Hundley 11, Interlude remark (1)}).

\hypertarget{functional_equation}{}\subsubsection*{{Functional equation}}\label{functional_equation}

The [[functional equation]] of the Dedekind zeta function follows from its [[adelic integral]] representation in direct analogy to how this works for the [[Riemann zeta function]]. This is due to (\hyperlink{Tate50}{Tate 50}), highlighted by [[Ivan Fesenko]] in (\hyperlink{GoldfeldHundley11}{Goldfeld-Hundley 11, Interlude remark (1)}).

\hypertarget{the_pole_and_the_class_number_formula}{}\subsubsection*{{The pole and the class number formula}}\label{the_pole_and_the_class_number_formula}

The Dedekind zeta function $\zeta_K$ of $K$ has a [[simple pole]] at $s = 1$. The \emph{[[class number formula]]} says that its [[residue]] there is proportional the the product of the [[regulator of a number field|regulator]] of $K$ with the [[class number]] of $K$

\begin{displaymath}
\underset{s\to 1}{\lim} (s-1) \zeta_K(s)
  \propto
  ClassNumber_K \cdot Regulator_K
  \,.
\end{displaymath}
\hypertarget{RelationToThetaFunctions}{}\subsubsection*{{Relation to other zeta-, theta-, and L-functions}}\label{RelationToThetaFunctions}

\begin{itemize}%
\item The Dedekind zeta function $\zeta_K$ is the [[Artin L-function]] $L_{K,\sigma}$ for [[trivial representation|trivial]] [[Galois representation]]

\begin{displaymath}
\zeta_K = L_{K,1}
  \,.
\end{displaymath}
See at \emph{\href{Artin+L-function#RelationToDedekindZeta}{Artin L-function -- Relation with Dedekind zeta function}}.


\item The Dedekind zeta function has an expression as an integral over a kernel given by a [[Hecke theta function]] (\hyperlink{Kowalski}{Kowalski (2.3) (2.4)}).


\item The Dedekind zeta function of $K$ is equivalently the [[Hasse-Weil zeta function]] of $Spec(\mathcal{O}_K)$.



\end{itemize}
\hypertarget{analogs_over_complex_curves}{}\subsubsection*{{Analogs over complex curves}}\label{analogs_over_complex_curves}

The [[function field analogy]] in view of the discussion at \emph{[[zeta function of an elliptic differential operator]]} says that the Dedekind zeta function is [[analogy|analogous]] to the regulated [[functional determinant|functional trace]] of a would-be ``[[Dirac operator]] on [[Spec(Z)]]''.

[[!include zeta-functions and eta-functions and theta-functions and L-functions -- table]]

Other identifications/analogies of the Riemann zeta function (and more generally the [[Dedekind zeta-function]]) with [[partition functions]] in [[physics]] have been proposed, in particular the \emph{[[Bost-Connes system]]}.

[[!include zeta-functions and eta-functions and theta-functions and L-functions -- table]]

\hypertarget{function_field_analogy}{}\subsubsection*{{Function field analogy}}\label{function_field_analogy}

[[!include function field analogy -- table]]

\hypertarget{references}{}\subsection*{{References}}\label{references}

\begin{itemize}%
\item [[John Tate]], \emph{Fourier analysis in number fields, and Hecke's zeta-functions}, Algebraic Number Theory (Proc. Instructional Conf., Brighton, 1965), Thompson, Washington, D.C., pp. 305--34 1950


\item Wikipedia, \emph{\href{http://en.wikipedia.org/wiki/Dedekind_zeta_function}{Dedekind zeta function}}


\item E. Kowalski, section 1.4 of \emph{Automorphic forms, L-functions and number theory (March 12--16) Three Introductory lectures} (\href{http://www.math.ethz.ch/~kowalski/lectures.pdf}{pdf})


\item [[Dorian Goldfeld]], [[Joseph Hundley]], chapter 2 of \emph{Automorphic representations and L-functions for the general linear group}, Cambridge Studies in Advanced Mathematics 129, 2011 (\href{https://www.maths.nottingham.ac.uk/personal/ibf/text/gl2.pdf}{pdf})



\end{itemize}
[[!redirects Dedekind zeta functions]]

[[!redirects Dedekind zeta-function]] [[!redirects Dedekind zeta-functions]]



\end{document}